Question

Let \(X\) be a random variable having the moment generating function

• A. \(16\)
• B. \(4\)
• C. \(6\)
• D. \(8\)

Hint:

For a Poisson distribution, the mean and variance are both equal to the parameter \(\lambda\).

Answer 1

The Correct Option is D

Step 1: Recognise the distribution.
The MGF $M_X(t)=e^{2(e^t-1)}$ matches the Poisson form $e^{\lambda(e^t-1)}$ with $\lambda=2$.

Step 2: Read the variance.
For Poisson, the variance equals $\lambda$, so $\mathrm{Var}(X)=2$.

Step 3: Scale it.
Adding a constant changes nothing and multiplying scales by the square: $\mathrm{Var}(2X+3)=4\,\mathrm{Var}(X)$.

Step 4: Compute.
$4\times2=8$.

Step 5: Conclude.
This is option (D).
\[ \boxed{8} \]